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Mirrors > Home > ILE Home > Th. List > zaddcllemneg | Unicode version |
Description: Lemma for zaddcl 9062. Special case in which is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zaddcllemneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 967 | . . . . 5 | |
2 | 1 | recnd 7762 | . . . 4 |
3 | 2 | negnegd 8032 | . . 3 |
4 | 3 | oveq2d 5758 | . 2 |
5 | negeq 7923 | . . . . . . . 8 | |
6 | 5 | oveq2d 5758 | . . . . . . 7 |
7 | 6 | eleq1d 2186 | . . . . . 6 |
8 | 7 | imbi2d 229 | . . . . 5 |
9 | negeq 7923 | . . . . . . . 8 | |
10 | 9 | oveq2d 5758 | . . . . . . 7 |
11 | 10 | eleq1d 2186 | . . . . . 6 |
12 | 11 | imbi2d 229 | . . . . 5 |
13 | negeq 7923 | . . . . . . . 8 | |
14 | 13 | oveq2d 5758 | . . . . . . 7 |
15 | 14 | eleq1d 2186 | . . . . . 6 |
16 | 15 | imbi2d 229 | . . . . 5 |
17 | negeq 7923 | . . . . . . . 8 | |
18 | 17 | oveq2d 5758 | . . . . . . 7 |
19 | 18 | eleq1d 2186 | . . . . . 6 |
20 | 19 | imbi2d 229 | . . . . 5 |
21 | zcn 9027 | . . . . . . . 8 | |
22 | 21 | adantr 274 | . . . . . . 7 |
23 | 1cnd 7750 | . . . . . . 7 | |
24 | 22, 23 | negsubd 8047 | . . . . . 6 |
25 | peano2zm 9060 | . . . . . . 7 | |
26 | 25 | adantr 274 | . . . . . 6 |
27 | 24, 26 | eqeltrd 2194 | . . . . 5 |
28 | nncn 8696 | . . . . . . . . . . 11 | |
29 | 28 | ad2antrr 479 | . . . . . . . . . 10 |
30 | 1cnd 7750 | . . . . . . . . . 10 | |
31 | 29, 30 | negdi2d 8055 | . . . . . . . . 9 |
32 | 31 | oveq2d 5758 | . . . . . . . 8 |
33 | 22 | ad2antlr 480 | . . . . . . . . . 10 |
34 | 29 | negcld 8028 | . . . . . . . . . 10 |
35 | 33, 34, 30 | addsubassd 8061 | . . . . . . . . 9 |
36 | peano2zm 9060 | . . . . . . . . . 10 | |
37 | 36 | adantl 275 | . . . . . . . . 9 |
38 | 35, 37 | eqeltrrd 2195 | . . . . . . . 8 |
39 | 32, 38 | eqeltrd 2194 | . . . . . . 7 |
40 | 39 | exp31 361 | . . . . . 6 |
41 | 40 | a2d 26 | . . . . 5 |
42 | 8, 12, 16, 20, 27, 41 | nnind 8704 | . . . 4 |
43 | 42 | impcom 124 | . . 3 |
44 | 43 | 3impa 1161 | . 2 |
45 | 4, 44 | eqeltrrd 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 cr 7587 c1 7589 caddc 7591 cmin 7901 cneg 7902 cn 8688 cz 9022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: zaddcl 9062 |
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